The Sun is 99.9% of the mass in our solar system. Jupiter is most of the remaining 0.1% and then the rest of the planets are included in the 0.01%.
Also, Jupiter is the only planet that orbits around a point outside of the surface of the Sun. All the other planets point of orbit are below the surface of the Sun.
Every two body orbit involves a barycenter that is not exactly in the center of either body (i.e. every planet/sun combo "orbit around one another").
The unique aspect of Jupiter is that the Jupiter/Sun barycenter is outside of the surface of the Sun. This requires an incredible amount of mass in Jupiter.
What if we demote the sun? It is in a class of stars that is too small to have a internal barycenter for all it's planets, there are larger stars that do because planets like Jupiter have a limit before they start being classed as Brown Dwarfs.
The sun is a pretty ordinary star. It's not nearly so small that it should be demoted or we would be rewriting our entire stellar classification system.
74% (don't quote me here, this is from memory) of stars are less massive red dwarfs. Only about 4% (again don't quote) stars fall into our weight class.
Yes but I mean our sun is not at the low end of that spectrum. As you say, most stars are smaller than the sun. So rewriting the system would mean that the majority of stars are no longer stars.
We're not just rewriting the system for the sun though. There have to be many stars that aren't large enough to have internal barycenters for all their planets. Maybe we do want to define the stars that have this trait and not the planets in their solar systems.
Also, Jupiter is the only planet that orbits around a point outside of the surface of the Sun. All the other planets point of orbit are below the surface of the Sun.
Jupiter is massive.